Curve deform - Follow along a path
If you want to use an armature, follow the Batfinger's tuto here.

If you want to use a simple Curve modifier, you need to make sure that the position of your object on the curve. What you should do is:


Choose a vertex in your curve, where your animation is supposed to begin.
Make it the curve's origin.
Make it the curve’s first point. To do so, enable the Bsurfaces addon, select the vertex, in the Tools panel > Tools > Bsurfaces > click Set First Points.
Now put your dolphin object at the exact same point.
Move it on the right axis so that it follows the curve.